Properties

Degree $2$
Conductor $7056$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·11-s − 4·13-s − 6·17-s − 8·19-s − 6·23-s − 25-s + 10·29-s − 4·31-s + 6·37-s + 6·41-s − 4·43-s + 8·47-s − 2·53-s − 4·55-s − 4·59-s − 8·61-s + 8·65-s + 8·67-s − 10·71-s + 4·73-s − 4·79-s + 12·83-s + 12·85-s + 14·89-s + 16·95-s + 4·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.603·11-s − 1.10·13-s − 1.45·17-s − 1.83·19-s − 1.25·23-s − 1/5·25-s + 1.85·29-s − 0.718·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s − 0.274·53-s − 0.539·55-s − 0.520·59-s − 1.02·61-s + 0.992·65-s + 0.977·67-s − 1.18·71-s + 0.468·73-s − 0.450·79-s + 1.31·83-s + 1.30·85-s + 1.48·89-s + 1.64·95-s + 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{7056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8116985192\)
\(L(\frac12)\) \(\approx\) \(0.8116985192\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.899113846890409665577011513332, −7.33617816455226293213945347049, −6.43760935956942238381405409471, −6.14421070646052528445807958869, −4.77041920554517839461286789177, −4.37455115973192812140463707191, −3.79766730312154259979498127992, −2.59150709519871201789294173801, −1.98176544333354037351104687406, −0.43333167368032222610975469437, 0.43333167368032222610975469437, 1.98176544333354037351104687406, 2.59150709519871201789294173801, 3.79766730312154259979498127992, 4.37455115973192812140463707191, 4.77041920554517839461286789177, 6.14421070646052528445807958869, 6.43760935956942238381405409471, 7.33617816455226293213945347049, 7.899113846890409665577011513332

Graph of the $Z$-function along the critical line